if $A$ subset of $B$ then $P(B)-P(A)=P(B\setminus A)\ge 0$

probability real-analysis probability-theory

$P(B)-P(A) = P(B \cap A) + P(B \cap A^c) - P(A \cap B) - P(A \cap B^c)$,

but $P(B \cap A) = P(A \cap B)$, and since $A \subset B$ we have $P(A \cap B^c)=P(\emptyset )=0$.

So $P(B)-P(A) = P(B \cap A^c)$ which is $P(B \backslash A)$.

$P(B\backslash A)\ge 0$ comes from the definition of a probability measure.

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